MathDB

Let

\varepsilon < \frac{1}{2}

be a positive real number and let

U_{\varepsilon}

denote the set of real numbers that differ from their nearest integer by at most

\varepsilon

. Prove that there exists a positive integer

m

such that for any real number

x

, the sets

\left\{x, 2x, 3x, . . . , mx\right\}

and

U_{\varepsilon}

have at least one element in common.
proposed by the ICMC Problem Committee

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